Integrand size = 20, antiderivative size = 296 \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b c e x^2 \sqrt {-1+c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {2 b x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
1/3*x*(a+b*arcsec(c*x))/d/(e*x^2+d)^(3/2)+2/3*x*(a+b*arcsec(c*x))/d^2/(e*x ^2+d)^(1/2)+1/3*b*c*e*x^2*(c^2*x^2-1)^(1/2)/d^2/(c^2*d+e)/(c^2*x^2)^(1/2)/ (e*x^2+d)^(1/2)-1/3*b*c^2*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^( 1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*d+e)/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e* x^2/d)^(1/2)-2/3*b*x*EllipticF(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1 +e*x^2/d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 4.13 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {x \left (b c e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )+a \left (c^2 d+e\right ) \left (3 d+2 e x^2\right )+b \left (c^2 d+e\right ) \left (3 d+2 e x^2\right ) \sec ^{-1}(c x)\right )}{3 d^2 \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}-\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+2 \left (c^2 d+e\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{3 \sqrt {-c^2} d^2 \left (c^2 d+e\right ) \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
(x*(b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) + a*(c^2*d + e)*(3*d + 2*e*x ^2) + b*(c^2*d + e)*(3*d + 2*e*x^2)*ArcSec[c*x]))/(3*d^2*(c^2*d + e)*(d + e*x^2)^(3/2)) - ((I/3)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^ 2*d*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] + 2*(c^2*d + e)*Ellip ticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*(c^2*d + e)* Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])
Time = 0.53 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.92, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5751, 27, 402, 27, 399, 323, 323, 321, 331, 330, 327}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5751 |
| \(\displaystyle -\frac {b c x \int \frac {2 e x^2+3 d}{3 d^2 \sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx}{\sqrt {c^2 x^2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \int \frac {2 e x^2+3 d}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )^{3/2}}dx}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {b c x \left (\frac {\int \frac {d \left (e x^2 c^2+3 d c^2+2 e\right )}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{d \left (c^2 d+e\right )}-\frac {e x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \left (\frac {\int \frac {e x^2 c^2+3 d c^2+2 e}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{c^2 d+e}-\frac {e x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle -\frac {b c x \left (\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx+2 \left (c^2 d+e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{c^2 d+e}-\frac {e x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle -\frac {b c x \left (\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx+\frac {2 \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}dx}{\sqrt {d+e x^2}}}{c^2 d+e}-\frac {e x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle -\frac {b c x \left (\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx+\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{\sqrt {c^2 x^2-1} \sqrt {d+e x^2}}}{c^2 d+e}-\frac {e x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {b c x \left (\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx+\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}}{c^2 d+e}-\frac {e x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle -\frac {b c x \left (\frac {\frac {c^2 \sqrt {1-c^2 x^2} \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{\sqrt {c^2 x^2-1}}+\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}}{c^2 d+e}-\frac {e x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle -\frac {b c x \left (\frac {\frac {c^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{\sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}}{c^2 d+e}-\frac {e x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {c^2 x^2}}+\frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sec ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {b c x \left (\frac {\frac {2 \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {c \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{\sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}}{c^2 d+e}-\frac {e x \sqrt {c^2 x^2-1}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {c^2 x^2}}\) |
(x*(a + b*ArcSec[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcSec[c*x]) )/(3*d^2*Sqrt[d + e*x^2]) - (b*c*x*(-((e*x*Sqrt[-1 + c^2*x^2])/((c^2*d + e )*Sqrt[d + e*x^2])) + ((c*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcS in[c*x], -(e/(c^2*d))])/(Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (2*(c^2 *d + e)*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/( c^2*d))])/(c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]))/(c^2*d + e)))/(3*d^2*Sqr t[c^2*x^2])
3.2.59.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSec[c*x]) u, x] - Simp[b*c*(x/Sqrt[c^2*x^2]) Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1 /2, 0])
\[\int \frac {a +b \,\operatorname {arcsec}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Time = 0.13 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.18 \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {{\left (2 \, {\left (a c^{3} d^{2} e + a c d e^{2}\right )} x^{3} + 3 \, {\left (a c^{3} d^{3} + a c d^{2} e\right )} x + {\left (2 \, {\left (b c^{3} d^{2} e + b c d e^{2}\right )} x^{3} + 3 \, {\left (b c^{3} d^{3} + b c d^{2} e\right )} x\right )} \operatorname {arcsec}\left (c x\right ) + {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} + {\left ({\left (b c^{4} d e^{2} x^{4} + 2 \, b c^{4} d^{2} e x^{2} + b c^{4} d^{3}\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left ({\left ({\left (b c^{4} - 3 \, b c^{2}\right )} d e^{2} - 2 \, b e^{3}\right )} x^{4} + {\left (b c^{4} - 3 \, b c^{2}\right )} d^{3} - 2 \, b d^{2} e + 2 \, {\left ({\left (b c^{4} - 3 \, b c^{2}\right )} d^{2} e - 2 \, b d e^{2}\right )} x^{2}\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{3 \, {\left (c^{3} d^{6} + c d^{5} e + {\left (c^{3} d^{4} e^{2} + c d^{3} e^{3}\right )} x^{4} + 2 \, {\left (c^{3} d^{5} e + c d^{4} e^{2}\right )} x^{2}\right )}} \]
1/3*((2*(a*c^3*d^2*e + a*c*d*e^2)*x^3 + 3*(a*c^3*d^3 + a*c*d^2*e)*x + (2*( b*c^3*d^2*e + b*c*d*e^2)*x^3 + 3*(b*c^3*d^3 + b*c*d^2*e)*x)*arcsec(c*x) + (b*c*d*e^2*x^3 + b*c*d^2*e*x)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d) + ((b*c^4 *d*e^2*x^4 + 2*b*c^4*d^2*e*x^2 + b*c^4*d^3)*elliptic_e(arcsin(c*x), -e/(c^ 2*d)) - (((b*c^4 - 3*b*c^2)*d*e^2 - 2*b*e^3)*x^4 + (b*c^4 - 3*b*c^2)*d^3 - 2*b*d^2*e + 2*((b*c^4 - 3*b*c^2)*d^2*e - 2*b*d*e^2)*x^2)*elliptic_f(arcsi n(c*x), -e/(c^2*d)))*sqrt(-d))/(c^3*d^6 + c*d^5*e + (c^3*d^4*e^2 + c*d^3*e ^3)*x^4 + 2*(c^3*d^5*e + c*d^4*e^2)*x^2)
Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
1/3*a*(2*x/(sqrt(e*x^2 + d)*d^2) + x/((e*x^2 + d)^(3/2)*d)) + b*integrate( arctan(sqrt(c*x + 1)*sqrt(c*x - 1))/((e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e*x^ 2 + d)), x)
\[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]